# Simple harmonic motion

A2

A body moving with simple harmonic motion satisfies two conditions:

1. Acceleration is directly proportional to displacement from the equilibrium position and is
2. Directed towards that equilibrium position

The body moves in a straight line through the equilibrium position, and the acceleration is caused by a restoring force.

The motion therefore follows these equations:

f is given a name, the angular frequency, with symbol ω (greek omega):

Examples of systems:

Mass on a spring:

We can resolve the forces:

And solve by equating with the original equation for acceleration and rearranging (remembering that T = 1 / f):

Simple pendulum:

We can resolve the forces again, and use the small angle approximation, the definition sinθ = opposite side / adjacent side and the equation for acceleration:

This only works when the small angle approximation is valid, so angles must be less than 5°.

Damping

Damping is when oscillations reduce in amplitude due to the presence of resistive forces like friction and drag. These resistive forces are called dissipative forces.

There are three types of damping:

1. Light damping: the amplitude of oscillations decreases gradually
2. Heavy damping: the system returns to equilibrium slowly without oscillating (when damping is very strong)
3. Critical damping: the system returns to equilibrium in the least possible time without oscillating

Resonance

If an external force is applied to an oscillation and it changes the frequency of the oscillation, the oscillation becomes force. If the applied force is the same frequency as the original oscillation, resonance occurs. This means that the amplitude becomes very large.

Energy

The total energy is constant, so maximum potential energy = maximum kinetic energy. Both vary sinusoidally:

The maximum potential energy occurs at maximum displacement, when the object is changing direction so momentarily has zero velocity (and zero kinetic energy). This is when x = A, so maximum potential energy, and therefore total energy at any time is given by:

University

The equations for simple harmonic motion can be derived from those for circular motion:

There may also be phase differences between systems, denoted by φ (greek letter phi), so the displacement is written:

and similarly for velocity and acceleration.

is the angle with the x axis at any time, t.

Damping

Damping force is given the symbol b. As damping increases, the amplitude decreases and the time period increases.

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